The asymmetry indices of stable measures
Let \(\alpha\in (0,2]\), X be a locally convex space and \(X^*\) its dual. For every \(f\in X^*\) and every measure \(\Gamma\geq 0\) on X with bounded support let \(Q(\alpha,\Gamma,f) = tg(\pi \alpha /2)\int f| f|^{\alpha -1}d\Gamma\) for \(\alpha\neq 1\) and \(-2\pi^{-1}\int f \ln | f| d\Gamma\) for \(\alpha =1.\) The author introduces the concept of asymmetry index \(\beta(\mu)\) of a stable probability \(\mu\) on X, with exponent \(\alpha\). If \(X=R^ n\), \(\beta(\mu)\) is the infimum of all \(\beta\geq 0\) for which the characteristic function of \(\mu\) may be represented as \[ (1)\quad \phi_{\mu}(f)=\exp (if(a)-\int | f|^{\alpha}d\Gamma + i\beta Q(\alpha,\Gamma,f)), \] with an \(a\in X\) and some \(\Gamma\). For a general X, \(\beta(\mu)\) is the supremum of all \(\beta(\mu \circ T^{-1})\) with linear continuous \(T:X\to R^ n\), \(n=1,2,... \). We have \(\beta(\mu)\in [0,1].\) An open problem is: does \(\beta(\mu)\) change if we restrict to \(n=1?\) The author proves that \(\beta(\mu \circ T^{-1})\leq \beta(\mu)\) for a linear continuous T, with \(=\) if \(\ker T=0\), \(\beta(\otimes_{n\geq 1}\mu_ n) = \sup_{n}\beta(\mu_ n)\) and that \(\beta(\mu)\) is the absolute value of the ''usual coefficient'' for \(X=R\) and for the distribution on the function space of a homogeneous stable process with independent increments. Every stable \(\mu\) equals \(\mu_ 1*\nu\), where \(\mu_ 1\) is the symmetrisation of \(\mu\) transported by a c.1 and \(\nu\) is stable with \(\beta(\nu)=1\) if \(\beta(\mu)>0\) and degenerated if \(\beta(\mu)=0\). c is expressed in terms of \(\beta(\mu)\) and is extremal among all for which such a representation, without supposing \(\nu\) stable, exists. If for all stable probabilities on X the representation (1) is possible with \(\beta =1\) (particularily if X is quasicomplete), then the set of all \(\beta\geq 0\) for which this is possible is \([\beta(\mu),1].\)
stable probabilities on locally convex spaces
stable process with independent increments
scientific article; zbMATH DE number 4007317